Klein Bottle

In 1882, Felix Klein imagined sewing two Möbius Loops together to create a single sided bottle with no boundary. Its inside is its outside. It contains itself.

Take a rectangle and join one pair of opposite sides -- you'll now have a cylinder. Now join the other pair of sides with a half-twist. That last step isn't possible in our universe, sad to say. A true Klein Bottle requires 4-dimensions because the surface has to pass through itself without a hole.

It's closed and non-orientable, so a symbol on its surface can be slid around on it and reappear backwards at the same place.You can't do this trick on a sphere, doughnut, or pet ferret -- they're orientable.

A true Klein Bottle lives in 4-dimensions. But every tiny patch of the Klein Bottle is 2-dimensional. In this sense, a Klein Bottle is a 2-dimensional manifold which can only exist in 4-dimensions!

AH topology - seems interesting, but never really into it, probably because she was under huge pressure of qualifying exam with two famous Russian professors long time ago. God knows how she passed it :-)))

I'd love to have one of those and put a lady bug on it. Did you try on a Mobius loop?

This place to buy one of those:  http://kleinbottle.com/index.htm

  Klein Bottle Hat with a matching Mobius Scarf